A hybrid of two theorems of Piatetski-Shapiro

Abstract

Let $c>1$ and $0<\gamma <1$ be real, with $c\notin \mathbb{N}$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1,p_2,\dots ,p_s$ of index $\gamma$---that is, primes of the form $p=\lfloor m^{1/\gamma }\rfloor$ for some $m\in \mathbb{N}$.